 If you go to pitmath.com, you will see under block, if you click on math 12 links and where I have the video links, you've all navigated your way there before, under block D, under block B, and under block H by later this afternoon, you will already see, I think it says trig tutorial as the video, but also this morning in my block Bs, I spent the whole 80 minutes doing nothing but question and answer, that's already there under block B, I think I called it block B trig review, I'll stick it under block D and under block H, but it's there, if it doesn't get under block D and H, click on block B, it's there. So that's on the page where it has three columns, block B, D and H, where the lessons are. I haven't uploaded today's stuff yet as PDF files to my sky drive, that may or may not happen, we'll find out. So questions either from the review or from the identities review, let me know where it's from and we'll go. And you probably want your formula sheet in front of you, it'll help, right? But no hints sheet, so you have the formula sheet, you're bringing that into the test, but you're not bringing in the hints sheet, that you kind of got to know. And yeah, we're good, D and I had one, identities or review, okay, one number, a word problem. Hey, let's jump to the written because there was a bunch on the written at the end. 21 is almost identical to the quiz, so I'm going to pass, let's find a different one. 21, let's see, show me a word problem, show me, here we go, 10, number 10, okay, a Ferris wheel has a radius of 25 meters and its center is 27 meters above the ground. It rotates once every 40 seconds. I think that's the period, right, it's going to be 2 pi over 40 and I'm always trying to identify stuff. What's the radius? I always do a sketch though. So here's my high-tech sketch and I said I want to find highest, lowest and middle. Now this question is a bit unusual, they told me the middle, where's the center? I'm going to go jump. Then what's my amplitude? Well, if my radius is 25, that's my amplitude. That means I go 25 down to 2 and 25 up to 52. Yes? Which gives me some of my equation for part A when it says write the equation. My vertical displacement is the middle and my amplitude is the radius. Is that okay so far, Dino? Okay, now on some of the other ones they didn't give you the center and said they gave you the highest and lowest, but what I said is you're trying to find highest, lowest, middle in whatever order and if you know some you can find the other ones. If I knew the highest and lowest I could add them up, divide by 2, that would give me the middle and then I could subtract to figure out the amplitude. Is that okay? Then they will always give me one point somehow. In here it's this phrase, Sandy gets on the ferris wheel, where? She gets on or he gets on right there. So I would say to myself, self, which graph starts at the bottom? Positive sign, negative sign, positive coast or negative coast? The easiest here will be negative coast. Negative coast, because that will give me a phase shift of nothing. What was the period? Because I'm starting on the y-axis. If they had said, after three seconds Sandy gets on the lowest point, it would be T minus 3, the negative coast. Okay, yeah, period. No silly guesses, we're going to concentrate, we're going to think. Just what's the period? You're giving me B, which we're going to get to. What's the period? How long to go around once? You're making it too tough. It's right there. Read it. How long to go around once? Okay, what's it ready? What's the period? 40. What's B? 2 pi over 40, which would be, yeah, pi over 20, I think they probably, but I don't care, it's written. It's multiple choice, all reduced, and of course, my answer is to pick from, it's written, I'm not going to care. There's the equation. That means, by the way, after 40 seconds, we'd be back here again. For what it's worth, after 20 seconds, where would we be? There. So if you wanted to write it as a positive coast, or for some reason they insisted, write it as a positive coast, positive 25 coasts, same period. T minus 20 plus 27. What if they wanted to write it as a positive sign? I think positive sign, T minus 10, there's one in the middle. What if they wanted to write it as a negative sign? Oh, only someone silly would do it. Anyways, I do negative coasts. I don't know. What's my phase shift? I'm on the y-axis, so what's my phase shift? Yeah. Right? That's why we picked negative. Negative coasts would be by far the easiest one to pick. But you could do anyone you want to. I always go with the easiest, but make sure I show you, just in case, for some reason, the question on the provincial insists you use a certain trig function. Fine, I could move to wherever I need to. B says, determine the first time T. I think they're asking me to find an x when Sandy will be 35 meters. Hey, that's a y value. They're giving me a y value and asking me to find an x value. Now, regardless, I'm going to my graphing calculator. I'm carefully typing this in. Negative 25 coasts to pi x over 40 plus 27. View window. x is this. I think from 0 to 40 would probably make sense. Good scale. You know what? Since I noticed my dots are every 10, because it's a fairly easy one to spot, I think I'd go 10. Usually, I'm not too fussy, but that's convenient. Wise, I like to see the ground. How high did we go? 52. What's a nice number a bit higher than that so that we don't accidentally go off the screen? 55? 60? Oh, 60. Sure, let's go 60. Good scale there. 10. If we've done our play right, it should be Ferris Wheel goes up. Ferris Wheel goes down, and it should be exactly one period. And I could double check to make sure I'm right, because I could say, in my sketch at 20, I'm 52 high. If I go trace 20, I am 52 high. You know what? I'm pretty sure I haven't fluked the right equation. Now let's answer B. B, they said, when they're 35 meters high. That's a y value, so I'm going to go y2, 35. Wrap, wrap. What am I looking for? Where are these cross? Second function, calculate, intersection. First curve, second curve. Use the arrow keys to get closer. Good enough. Yes. It looks like t equals, did they tell me how many decimals they didn't? I'll go 12.1 seconds. What I say, you can solve this by hand. So you put a 35 there. Minus 27 from both sides. You divide by negative 25. I have a period change. Replace this with an A. We've done period changes. Not one's this ugly. But yeah, I would say tell your tutor, no, this one you're allowed to, and the exam specs are, say, you're allowed to use a graphing calculator to solve. In other words, in terms of work, the work to me was getting this. Now the risk with these, unfortunately, Dina, is if you get this wrong, you're getting this wrong. Sorry. By the way, I'll make this either worth two marks or four marks because there's one, two, three phase shift. Four things you're finding. I don't know how the heck they marked that out of three. That would have been silly to me. I'm the written. Absolutely. Sure. Since I'm right here. Oh, another Ferris wheel. I'm going to read it. Blah, blah, blah, blah, blah, blah, blah, blah, blah, blah. You know what? I got to do a sketch. They want me to get the equation. I have to do a little bit of a sketch here. So I'm interested in highest, lowest, and middle. Oh, they gave me the middle again. See it? I see the word center. So I would go, I'll change colors. I would go 32. They were a little sneaky here. They didn't give me the radius. What did they give me instead? What's the radius? In other words, 30 up and 30 down from the middle. 62 and 2. So if I'm finding the equation H of t, if I want to use the fancy schmancy function notation, you don't have to. But I may as well because I'm an athlete. Doug, what's my amplitude? Yep. Sorry, say it again. No. Amplitude. 30. No, no, it's your radius. From here to here, here to here. 30, right? What's my vertical displacement? Most common mistake, by the way. Steph, I see kids do this. That's not your vertical, sorry? Question? I can re-explain it. You're asking about the what? So what's the middle? What's the radius of this? So you're going to go 30 up from the middle and 30 down from the middle. You're not touching the ground on a ferrous wheel unless something goes horribly wrong. And someone's losing their job if you do, let me assure you. A lot of kids for the vertical displacement, Jesse, do that. No, no, no, it's the middle of your graph, not the bottom of your graph. It's the middle of your graph. It's that. Now we need to find one point. Oh, gets on at its lowest. Now I've also seen them say, assume we start at top. That's a weird ferrous wheel, but I think in the review that I gave you, there is one where they've built like a loading platform and the kids actually get on in the middle or on top, whatever. Usually we get on at the bottom. I would do that. That tells me which trig function I'm going to use. Negative cost. What if I had a point right there? I'd use sine. Positive or negative? I don't know, depending on whether I'm going up or down. What if I had the point right there? I'd use positive cost. Whenever I can, I'd like no phase shift because it's sure nicer. So negative cost and no phase shift. To find B, I need the period. So B is going to be 2 pi over 48, pi over 24 in lowest terms. Pi over 24 in lowest terms, right? Top and bottom, divide by 2, divide by 2. It rotates once every 48 seconds. Don't overcomplicate the period. They'll tell you either how long to go around once or the most complicated they'll get is they'll say it has an RPM of 6. That means six times in one minute, six times in 60 seconds, 10 seconds. They'll give you a fairly easy conversion if they want to make it nastier. But almost always, they'll tell you how long to go around once. Or like the question I gave you on the quiz, they'll give you the highest and the lowest, which is exactly half a wave. That's half the period. And you double it to get the whole period. Those are the only two ways they sneak the period into a word problem. Am I asking RPM? No. Is it fair game on the provincial? Yeah. It just means that many times per minute. So per 60 seconds. Usually fairly easy math. Anyways, that's the equation. Kind of spread out. Sorry. But that's the fine, just to make Doug happy. There, that looks all pretty now. Could you do the graphing part? Are you OK on that? So B, I would carefully graph this in. Oh, period is 48. So I probably graph from 0 to 48, because that's when I'm back on the ground. Unless this question mentioned after 60 seconds, then I'd better make sure I graph up to at least 60. I probably graph to 70 or something like that. So I always glance at the question and see, is there a value that I need to go up to? If not, I do want periods worth. When in doubt. So I have to split it in four sections. I'm not sure what you mean. Oh, to graph it? So if I wanted to graph it then, I would say here, there's the period. What's that right there? You're right. Where will I be? Yep. So if we wanted to do this as a cosine equation, a positive cos would be that, T minus 24, phase shift of 24 to the right. That would be a positive cosine equation. Oh, by the way, what would that be right there? That's when you're apparently in the middle, because. And oh, what would that be right there? Yeah, I got to think a little bit. Say 36. Did you say 36? Good, good, good, good, good, good, good, good, good. There's one whole wave. Yeah. Do you really have to split that into two sections? Yeah, I like the ground zero, right? This, I started zero. And I'd go to probably 65 or probably 70, to be honest, just so I got some reading room in the top. I wouldn't, I don't like going right to 62, because I find on the graphing calculators, it chops a bit of that off. And I don't, I want to see that top, because that's one of the ones like I would, I would do a X scale of 12. And I would look to see that this was right above the second hash mark, then I know I'm right. Like then I'm like, no way I fluked into this. I'm pretty sure I got the right equation. Yay, I just got to be careful on my calculator and I should get this all right. Yep, one whole period. The only time it wouldn't be is if in here, in part B, it said, how high is he after 60 seconds? Then I'd say, oh, 60 is over here. I'm going to go from zero to 70 or, or to be honest, nerd that I am. I would go from zero to 96, two whole periods is what I would do, because I just try and think nice and symmetrically. But sometimes you'll need to go a bit further than one period if they ask you a time that's past one period. 36 multiple choice, four on the written. Doug, did I answer your question? Okay, the answers are there as you can check the answers, right? Sorry I haven't had a chance to make an answer key. I've gotten up to number 74 and that took me about four hours and then I just been dropped, ticked by a cold and flu. So I've been going home and doing the blanket couch thing. Sunday I went to the Olive Garden and had their unlimited soup and breadsticks for lunch. I just sat there and wearing my sweatshirt, my sweatpants and more soup, please. More free soup, please. More free soup, please. That was great. But it just wasn't conducive to doing math. Have you been to the Olive Garden unlimited? Oh, that Zippa, Tascana, that Italian soup that they all, I could guzzle that stuff. What number, kiddo? 36, six squared, three squared times two squared. Yeah, six squared, three squared times two squared. I'm asking you to do some math in your head there, mister. It's nine times four, okay? So 36, there is the cheating method, which is slower and then the math method. I'll do the math method first and then I'll show you why you don't hand this test in early, okay? The math method, clip, suggestions. I think so, I think so. So this is going to be, now, Martin, is the bottom of this fraction also going to be a fraction? Then I'm gonna make the top cos over one. I'd never want half fraction, like everything or nothing but never in-betweens. So I'm gonna write this as cos theta over one all over, cotangent is cos over sign, which technically you don't have to have memorized. You guys have noticed it is on your sheet, but I'm gonna suggest to you, if you're having to look up tangent is sine over cos sign, you haven't done enough of the homework. That's when kids just memorize because they're lazy. Plus, now cosy-cant is one over sign. What do you think one over cosy-cant is? It's sort of one over one over sign, which is actually just plain old sign. Oh, you see it? Well, we're gonna keep going anyways, you asked. And Martin, I would do that as well because fractions may have more. Here, this is a complex fraction, but are there any plus signs in here at all? I can still use my good old, how do I divide by fractions with the multiply trip? This is gonna be cos over one times sign over cos, and then the sine over one would drop down. And Martin, you already had your nerdy adrenaline rush. What did you see when you saw it? You said, hey, the cos is canceled, and this is really sine over one plus sine over one. And now I would argue the over ones don't make any difference. In fact, now I think they're worse. It's sine plus sine, which is just plain old. That one would be awkward because there is no actual, don't mix up this that is not the same as that, it's not. Here you're doubling the angle. Here you're doubling the decimal of the trig function when you went to your calculator. They are not interchangeable. In fact, sine two theta is two sine theta cos theta. If they want, Martin, in all honesty, if they really want to get nasty, they would do this. That's true, because if I divide the cos sign over, looking at that identity, if I get the two sine by itself, I think you would divide the cos sign over. I don't think they would, that would be if they'd gone through an entire provincial and they felt it was slightly too easy, and this was one of the questions they were asking, how can we tweak this? They would make that the answer to pick from, but. Because it says simplify, this is not simplified. This is the simplest simplified answer. They'd get, we would give them flak for that. We would criticize. And believe it or not, every once in a while, as math teachers or as physics teachers, we get questions tossed. We raise enough of a stink and we say, no, this is unclear or whatever. A part of it is also because the people who are printing the exams don't have a math background, so they may not realize that that too has to be up there as an exponent when they're doing all the fonts and the refaunting, I can be moved down. Things like that happen as well. I was on one of the committees for a while, for two years. And when we had one gout, she had done math 12. Oh, we got a lot done when she was doing the typing, but sometimes we'd have a different gout with clearly not done much past math 11. And so we had to explain everything and why. So it was the difference between going 10 squared theta and TAM subscripted to theta symbol. It was really faster the first way, the way it worked, you know, whatever. Two is still nice and we got it all done. But I know which one we proofread more. Four written? I'm gonna guess it might end up being sloppy with brackets. And since you guys are here, you guys get a hint. You ready for your test? On one of the identities, well, first I'll say it on all of them. If you're ever plugging in, oh, a sine squared, that's one minus cos squared. Or a secant, what is secant squared? It's on your formula sheet. Yeah, I don't have those ones memorized. One plus 10 squared. If you're ever plugging any of those, or any of the double angle identities in, put them in brackets just in case. You may not need the brackets, but there is one question where you're going to need them. Otherwise, you'll have a negative that won't vanish because you'll miss that there was a minus minus, which was a plus. You won't catch it if you don't put the brackets and I know from prior experience. So whenever you're substituting stuff in, just put it in brackets first and you might be able to get away without it. So Martin, you did this. Five marks, wow. Now, by the way, this one makes me nervous because I don't see lots of squareds in junk. The uglier they look, the easier they end up being because there's usually eight or nine different approaches. The cleaner they look to begin with, that's when I kind of go, e is probably only one or two ways that work. I do that and I would do a negative one over coast. Yes, secant is one over coast. And I'm done the right side. So in that hints sheet when I said, start with a more complicated side first, what I really meant now is spend most of your time on the complicated side, but if you have a really easy side, deal with it and chuck it. Suggestions, Martin? Yeah, this is going to be coast over sign. And since I just wrote a fraction, I'm absolutely putting the sign over one, minus one over sign. How many levels is this fraction now, Martin, my friend? This is a complex fraction. Drop everything and get rid of it. How? Look at each mini fraction. What's your common denominator? I think just plain old sign. So very carefully, I'm going to multiply the top by sign and the bottom by sign. But what I've told you to make your life easier is make that a four level fraction. Multiply a four level by a four level. I don't really need brackets there, but I'll put them there. I do need brackets there, so I'll get in the habit of putting them there no matter what. On the top, Jesse, do you notice the sign cancels? And I'm left with just plain old. On the bottom, in the first term, nothing cancels, I get sign squared. Oh, I know where the negative came from. Whoa, wrong button. Sorry, I meant to hit this one. Here, oh, the minus sign would just drop down like a domino. A sign would cancel, and I have just a plain old one. No, it's not. You ready? Look up. Did you see it? Did you see it? If sign squared plus cos squared equals one, Martin, can you rewrite this to get sign squared minus one by itself? How? What, what, what, what, what? You're gonna have to minus this to there and minus this to there, and you're gonna get that sign squared minus one is not equal to cos squared. What's it equal to? And I smile because I was wondering where the heck that negative was gonna come from. There it is. Now I'm not so offended by that five marks. That's sneaky. And by the way, 2004 that I marked in 2004, I marked one like this where we caught every kid who just magically made the negative just kind of, it suddenly just appeared, we caught him because we knew what to look for. This is cos theta negative cos squared, which is negative one over cos. So the kids that we knew that had done your initial mistake and then just kind of used the force, if we didn't see Martin this step, I'll let you look, sorry. If we didn't see this step, we took a whole mark off because we said, you know what, they canceled and then they just kind of went, there's negative coming from somewhere. There told me you clued in, you had done that. The fact, you know what, you know what, I said to you that's three equations. What I should have said is that's six equations, isn't it? Because I can also get the negative cos squared by itself or negative sign squared by itself or negative one by itself because negative one would be negative cos squared minus sine squared. Yeah, six equations. Okay, get a sine squared minus one by itself. You would minus this to this side. How would you move the cos squared over? No. Dina, I'm gonna argue that's grade nine and 10 alge. That I won't call tricky. Tricky is a lot of the fractional stuff. Well, but here's the beauty. Hopefully most of you, even if you didn't catch that, if you were in a rush, Dina and you did this, oh, hang on. If you were in a rush and you did this and you got to here and you went, hopefully you would start saying, I've missed a negative somewhere. I'd give you four out of five. I'd give you four out of five. No, I have higher standards than that for you. Again, I mean, identities are tough. The nice thing is Martin, it's almost impossible to get a zero. Like if you can replace everything with sine and cosine, I give marks for that. Show you, just look up for a second. Here is a provincial exam answer key. New course, provincials. Just to let you know how they gave up part marks on an identity, okay? There are lots of part marks to go around. It looks like they gave a half mark for replacing sine 2x with 2 sine x cos x. Looks like they gave it, oh. Remember I said to you, be careful with brackets. This is an example of one where you really wanted to be careful with brackets, I think, because here you have a minus sign that later on might become a minus, minus, that's where, when you're substituting, be careful. It looks like they gave a half mark for replacing sine with, or sorry, tan with sine over cosine. Oh, and cotangent with cos over sine. Now, I have taught you, Steph, to also put that over one. Of course, they don't in the type font because that's extra work, extra pay of it, but I would have bought it. Oh, and you got a half mark for writing that in terms of, so in other words, just for replacing everything with sine and cos, one mark, and being able to replace sine 2x with 2 sine x cos x, which is right from your sheet, a half mark, that's 1.5. Oh, and then noticing a GCF in the bottom, that apparently got you, you were two out of five just for doing that, and yeah, I don't think you really started to do the funky stuff yet. They're tough to get zero on, sine squared. I might do it a different way, I gotta look at the original. This is a pretty scooking one, pretty big one. Yeah, this is June 2008 alternate solution, clearing the complex fraction, so this is using my method, except I would never have you write cos over cos, I would have you write cos over one over cos over one, so you have four levels equals four levels, but there it is, and I think you'll notice, one, two, three, four, five, one, two, three, four, okay, it's not shorter than their other method, usually it is. So, what's the rule? Don't you dare leave an identity blank. That's about the only thing that'll take me off on this test, because I think every one of you knows the tangent is sine over cosine, and if you don't, it's on your sheet, right? It really is, and that sine two theta is two sine theta cos theta. That's on your sheet, cos two theta, yeah, you got three choices, and you better believe one of your identities is gonna have a cos two theta, and force you to pick one of the best of the three, but any one of them will get you there. Did you have another one, Martin? I can't remember, I didn't do both of yours. Next, written or multiple choice? Written, this one here, this one? This one, it's funny, this one is so easy that it's tough. It really, I'm dead serious, because my block B's asked me to do this one, too. And I went, oh, okay. There's a reason why I've taught you certain triggers or phrases. So, you ready? What kind of an equation is this? How do I know, okay, what's the first thing I do before I do anything? Did you say make it equal to zero? There's a reason I've taught you all that, because I'm hoping that when you get it looking like this, you would say, well, now I need to factor. What's the first thing that you always, always, always, ah, there's a method to Mr. Dewick's insanity and madness? Yes, there is. You see it, you see it? And now it's actually really easy to factor. This is nowhere near as tough as some of the other ones. And not only that, Leslie, I'm gonna get really nice roots. What are my roots? Now, for both of these, if I wasn't so sore-throated and hoarse and cold, I would right away be bellowing alarm bells screaming, because it's zeros and ones. They're on here somewhere. The cast rule is not going to work. So, instead, we could either use the unit circle or, you know what, I'm gonna sketch cosine, because that's what we used also. Yeah, remember last year when you had an X by itself in front, one of your roots was zero. Okay, where is cosine one high? Here, the answer to this root is gonna be zero. And two pi, except I noticed, oh, they want me to give over the real numbers. They didn't give me a domain restriction. That means they want the general solution. You know how I know? Does it say between zero and two pi? Nope. When it says over the set of real numbers, that's also another way of saying the general solution, that I know there's no zero and two pi, they must want the general solution. Where is cosine zero high? Oh, that's two pi, that's pi, that's zero high, that's zero high. What are those values? Yep, and I would call this X1, X2, X3, but that's not the general solution. The general solution says take these and add the period. Do you see anything in front of the X's? Any number, any coefficient in front of the X's? In other words, do you see a B? No, so what's my period of cosine then? Two pi, oh, you know what? I'm gonna be clever. Remember how we wrote this? Come on. And then we said plus where is cosine one high? Right there, how did I get one? What are the roots? You gave me the root. Where is cosine one high? Right there, what X value is that right there, right? Also right there at two pi, and at four pi, and at six pi, and at eight pi, but I'm off my screen now, right? You see what I meant by that was so easy that it was tough? Because I have to be honest, I couldn't ask for nicer factoring or nicer roots, but spotting that, what's the most common mistake? Kids do this, and they somehow cancel a cos, and they get a one and they've lost a whole root. There is a reason that I've started every one of these with what kind of an equation is this? It's a quadratic, don't you dare start doing anything, make it equal to zero, right? Yep, we can get exact values from those guys, there they are. Next, 28? Written or multiple choice? Is there 28? Oh, it's 24 multiple, written, so now I know. 28, I don't know, maybe it is, maybe it isn't. Oh, another Ferris wheel. Okay, so here's the one I was talking about, Doug, where for some reason, I think we're gonna start out not on the ground, we're gonna start out in the middle. Okay, by the way, I knew that anyways, because I glanced at all four answers and I noticed they're all signed, so we'd better be starting in the middle. You didn't glance at all four answers, I just saw you look now. Oh, Dina, what have I taught you about multiple choice? Always look at the answers too, always, always. Because I think we can actually get away with only doing about two thirds of this question, watch. I said to you I'm interested in highest, lowest, and middle. Read the question very carefully, did they tell me the middle? Ah, so what's my vertical displacement? Look up, I'd be doing this, oh, hang on, I'm not with an eraser, I'd be doing that right away, because that hasn't got a vertical displacement, only those two do, right? Dina should start reading the answers right away and glancing, yes! And you know what, do these both have the same amplitude? I'm not gonna waste my time trying to figure out the amplitude, because apparently it's not gonna help me. And they both have the same phase shift, you know what, all I'm gonna worry about is figuring out what B is, and B is two pi over the period. Read the question carefully, how long to go around once, what's the period? So two pi over 32, and I gotta be a little careful, they're trying to suck me in here because what is that in lowest terms? See, ah, they put the 32 there to really try and snag me, or just slide dogs, I'm not falling for that, it's A. You see what I mean by, we really only did about two-thirds of that question, half of two, scarce. What if that was a written? You want me to do this as it was a written? No problem, no problem. I would do a sketch, right? I would do a sketch. I want to find highest, lowest, and middle. This one told me the middle. You said the middle was 20. Did they tell you the radius, because that's my amplitude? The radius always will be, the diameter is always twice, yeah. Think about, because really what you're doing is you're going that high and that low, and that happens to be the radius of a circle, so yeah. So amplitude is 18, which would mean 38 and two, and I would go, the vertical displacement is 20, and the amplitude is 18. I need one point. Well, this one says, a platform allows a passenger to get on the wheel at point P at the middle. It says this time we're starting out there. Which trig function starts in the middle? Now positive or negative, I'm not sure, to be quite honest. Oh, wait a minute, the diagram says they get on here and they go up or down? Up. So it's gonna be going up. Which one starts in the middle and goes up? Positive sign. Oh, and since I'm right on the y-axis, no phase shift. Two pi over 32. How long to go around once? So that means after 32 seconds, I'd be back here. What would that be? What would that be? Come on, easy math. You say eight, I hope. What would that be? And I think it would be like this. Starting here, we're going up. Middle, bottom, middle. So what if they had said, for some strange reason, write a positive cosine? Leave it, cos, leave it, t minus, yep, plus 20. What if they had said, for some stupid reason, write a negative sign? Negative, leave it, leave it, leave it, t minus. Negative sign starts in the middle and goes down, right? Negative cos would be 24. Whatever they want, I can do. But again, if you have a choice, if they don't specifically say, if they give you a point on the y-axis, use that as your trig function, they'll face it way easier. Less typing, less chance of making a typing mistake on my calculator, less chance of making a sloppy math mistake on my calculator, all that good stuff. Martin, you got a question? Oh, Doug. Multiple choice? Okay, this is from last test, mostly, but it's good review. Did I sign it? Oh, this is actually what you asked me today. Except yours, I think, was a cosecant to four pi by three or something. It's the same idea. Did they give you the angle? Yes, they want an exact value. It's the same question, the same procedure. I would do this, Doug. Did they give you the angle? Okay. Five pi by six is right there, right? C, A, S, T. First thing, Doug, my friend, is tangent negative or positive. So, of course, I've glanced at my answers and I'll be saying, uh-uh, uh-uh, good. What's my reference angle? I would draw the triangle that has the pi by six in it. Which angle is pi by six? The bottom one or the top one? What is the tangent of that? Opposite over adjacent. What? Don't think so. That's a hypotenuse, not an opposite. Tangent is, psst, opposite over adjacent. It's tan, right? No, it's root three. One, no, pi by six. You just can't read my writing. It's pi by six is this angle down here. Pi by three is this angle up there. This is the 30 degree, the small one. Don't get those mixed up, because that can domino through a whole test. One student, last test, did that, cost that student about 14 marks. They got the pi by three and the pi by six mixed up. And they wrote it all nice on the top page of the front page. They wrote out everything. Sign is y over r, cos is x over r, tan is y over x. And then they did the triangle and they labeled the angles wrong. And I saw that when I started marking and I was just, oh no! Please tell me you redrew it. Please tell me you redrew it. No, the student used that as a reference the whole time. They fixed that and they had a b. Multiple, hey, don't kid yourselves. Multiple choice in math is tougher. I would argue multiple choice math is harder than any other subject because I'll think of all the likely wrong answers and they are there. Have I tried to give you multiple choice skills this year? Reading, glancing at the answers, process of elimination, crossing off wrong answers? Yeah, I know. You tripped on your brain. We're just happy you have one to trip on. I lose the pool. Next, I'd love to. Since we're in the 20s, nice. How am I gonna solve this? Tell me instantly, how'd you know that? Decimals, right? I can't solve that by hand. Unless you know calculus, you can't solve that by hand. Calculus you can using, anybody here in Calc? Okay, using Newton's iteration method which is actually nerdly cool. It's one of the many reasons we're impressed with Sir Isaac Newton is holy smokes. You gave us a method that can solve any equation. Wow, that came in handy before the computing days. Believe me. Oh, and it can solve it fairly quickly. Any equation, I can solve it about a three quarters of a page, which sounds like a lot, but considering the options, that was doable, right? So graphing calculator. You ready? So you're clear. Left side, sine three x plus. Now tan is gonna give me weird stuff. It's gonna give me those weird vertical lines that aren't really supposed to be there. I'm prepared for that. The right side, three. Not gonna hit graph yet because I gotta change my windows. Gotta type in window. Oh, did they give me a domain restriction? I'll go zero to two pi. And when in doubt, what's my scale? Pi by six, right? Why min? Well, the amplitude of sine is one. Tangent, though, is, right? Oh, and this is three high. So you know what? I better make sure the number three high appears on my graph. I'm gonna go from negative five to five scale one, just to see what that looks like. And if I needed to, I'd redo my window. But hit graph. Ugly cousin, okay? Now, Stephanie, is that line and that line really supposed to be there? No, how many answers are there? Count two, not four, okay? That's not a solution right there. And that's not a solution right there. Only Leslie goes to our solutions. So have you crossed off C and D? And if you haven't, do it right now, right? By the way, how much you wanna bet C and D include that thing, which isn't supposed to be there and that thing, which isn't, did I just do this with you earlier today in class? I'm trying to remember. Okay, good, same question. Yeah, intersection. And you know what? I think I'm only gonna find the first one, because I'm done. I'm not gonna bother finding the second one. It's a waste of my time. Second function, calculate intersection. First curve, second curve. Oh, okay, you guys ready for a shortcut? Look up. You'll notice when you're using intersection by default, it puts you in the middle. The answer is either 1.3 or 2.4. Rather than holding the arrow key down, I'm gonna make my guess one, which is close to both of them and hit enter. It's faster than holding the arrow key down. You can actually type in a number for a guess. I don't do it very often, but since I noticed both of those were close to each other, why not pick a number close to them, a nice, clean number, he'll find it. Less typing. A, yes? Okay? One of those questions. One of those questions. Not a quadratic. A solve by, solve by graphing. How do you know? That's my answer. Those asymptotes are your computer not knowing what to do. You have a TI-84 or a TI-83? Because the 84s are better and worse. The 84s, they change the software. Those lines don't appear, but you gotta be careful. Sometimes the computer will still try and tell you it crosses there. It'll stop just hovering in midair, which is almost worse. And I can't find, like, this one here. Let me see. If I go like this. Second function, calculate, intersection, first curve, second curve. I'm gonna go just to the right to see if it actually stops right there where it shouldn't. Okay, it didn't. I have had it stop there sometime. I don't know when it does and when it doesn't. So that's why I've taught you all. This is why we don't just give you all graphing calculator in grade eight. The software still is not perfected. You have to be able to interpret. Otherwise, most of what I'm teaching, like a lot of what I'm teaching you, they give you a hefty Skookum iPod in grade eight or whatever and you'd be fine. The software's all on there. No, there's limitations for the free stuff. The expensive stuff can work around it. Yeah, since we're in the 20s. Oh, how about I scroll up on here? This is a C plus turned into an A minus. I'll show you what I mean. I definitely said I've done enough of these that I'm seeing alpha plus beta and alpha minus beta and I'm gonna fill those out. So can you do me a favor, please, my friend? You have your formula sheet there? I'd like you, first we're gonna do alpha plus beta, sine alpha plus beta. Read it out to me, but instead of alpha, say X and instead of beta, say pi by three. No, no, no, you don't have to memorize it. Like it's on your sheet. Don't memorize it. This sine alpha plus beta, you're reading that to me? Yeah, I don't think it's coast, coast, sine, sine. Doug, you need to find that because that's what this is. Okay, read me the right side, my friend, but instead of alpha, say X and instead of beta, say pi by three, go. Yep, yep, and this second term here, my friend, is alpha minus beta. It's this one. Read it to me, but instead of alpha, say X and instead of beta, say pi by three, go. So Jesse, all I did here was, oh, it's two addition identities side by side. Substitute them carefully because the alphas and betas are easy to confuse and there's lots of signs. You know, be careful, but I got that. I don't know where I'm going yet, but I feel better, I've written something. You know, you can see it's two addition identities side by side. You see it, you see it, you see it, you see it. What cancels? There's a plus coast sign and there's a minus coast sign? Really? Oh, well that's good. I did that. That was my first step. And that left me with sine X coast pi by three plus sine X coast pi by three. You know what? That left me with two sine X coast pi by threes. Two. What don't you see in any of our answers anywhere that we still have kicking around on our question? Steph, what? I suppose this is how? Again, are we learning to look at the answers? There's lots of hints there. I said, you know, I think they want me to do something. Well, wait a minute. I have a triangle with the pi by three in it. It's the one, two, root three triangle. Which one is pi by three? The bottom one, Martin, or the top one, Martin? What is the coast sign of pi by three? A half? So are you saying that this whole thing is two times sine X times a half? See it? You know what two times a half is? I didn't add them. They canceled. No, no, no, no. So sine X is for the pi by three times the pi by three. So add that out of them. Okay, this is one weird term. Okay? This is like terms, though. This is like AB plus AB. But instead of an A, you know what I have? Okay, folks, keep your voices down. It's okay, I'm keeping the door open, whatever. This is AB plus AB. You know what AB plus AB is? Yeah, two ABs. Now let me show you a different way. Ready, Martin? Is this plus this? Does it equal that? Make it even easier. Let's use letters you're familiar with. What's XY plus XY? It's 2XY, it's not 2X2Y, it's 2XY. Now, just to convince you, here's what I could have done instead. I gathered like terms, supposing I hadn't. Supposing I had said, hey, what is the coast sign of pi by three? What is the coast sign of pi by three? You already told me what's the coast sign of pi by three. So if I hear you, this is a half sine X, I'll move the one half to the front, plus a half sine X, that's this one. You know what a half sine X plus a half sine X is? One sine X. Am I gonna write the one? No, still the same answer. That one's pretty tricky. What I think I said this was, if they had just given you one addition identity, that's C plus B, since you asked. Oh, the hardest question in this, there's about 10 curve balls here. Bear with me. Martin, I don't spend a bunch of time on the really, really nastiest, because they don't help you. There's gonna be like two on the multiple choice. Out of 14 or 15 or 16 questions, why would I spend time on one eighth of the multiple choice when the other 14 are the basic skills we need to practice to master? So, I mean, if you wanna know what some of the curve balls are, this is what I would say. 13 is tough. And what I mean by tough is it's a twist within a twist, or maybe even a twist within a twist within a twist, right? 27, which I think you guys just asked me. Give me a second here. 33, well no, well, if you... 33, I see cos cos minus sin sin. Look at your formula sheet. Do you have cos cos minus sin sin on there somewhere? Yeah, you read to me the left side of that, please. Oh, you know what alpha is? 3x, you know what beta is? 2x, it's cosine of 3x minus 2x. By the way, what is 3x minus 2x? 1x, it's cosine of x. If you see that, it falls apart. If you don't, if you try and actually somehow, if you don't see that this whole thing here is one big identity, well, you're not gonna get it. 37 is pretty tricky. 37. 47 is pretty tricky. Ah, well no, I've taken that back. 47, if you read it right, it's fine. I should tell you all the easy ones are tricky, and then when you try them and get them all right, you'd be like super confident. Ah, I should have thought of that. Oh well. 58, see, but all of these Martin are individual ones that never repeat. So, I mean, I can give you, you know, I'll keep going, and then I'll answer Doug's. I said 58, I think. 68, and you'll notice they're not very frequent. Oh, 75, I don't think I signed that, but it's nearly cool. Ooh, log y equals log of sine of x. Oh boy! No, you can't, because you can't do log y on your calculator. The y has to be by itself. They thought of that. Although, there's probably software you could download. 79, log cos, 88, 90 is an algebraic one. Kids find those tough. 95, which I think I assigned, and if I didn't, I showed up. That's a hint. 99 was cheap. 99, because he did the first answer key. But 99 is an x, y, and r question. It's from last year, and it's honestly so cheap that I almost found it offensive. So, this is from the 09 exam, but it was also out again. It came out about six years ago as well on one of the exams. 106, so I mean, it's kinda hit and miss. 115, but hopefully, if you look at 115 now, you would say, oh, that's an addition identity and another addition identity, or a subtraction identity. As soon as I see sine or cosine of something minus something or something plus something in brackets, I'm running off to those four alpha beta ones, right? Well, I've rambled. Which one did you want me to do? Love to, because telling you which ones are tougher is kinda to be a pointless exercise. Oh, okay, this is their way of getting around the graphing calculators. But just to help you show you how to do this, I'll show you how we would do it with numbers and then we'll see if we can generalize it. What would the maximum, I'm gonna make up some numbers for A and D and just rewrite it. What would the maximum value be if A was five and D was seven? How high does this graph go? What's the highest that graph gets? Y12. Vertical displacement is the middle. Seven is the middle. What's the amplitude? So from seven it goes five down, five up. Why is it C? Vertical displacement and then from there go A up. What if they wanted the minimum value? Which of those is the minimum value? B is the minimum value, right? There's an algebraic one and I think it was one of you two sitting over there. We did an algebraic one. I said, make up numbers when in doubt that fit the condition that they give you and you'll spot the pattern. Yeah, you, we were doing the ABC one, you and I, right Steph? It had the, it was, was it you? Number of, it was number of solutions and it had like A, A, make up numbers that match their condition. I know it's cold in here but it's keeping you guys awake and you're tired so we'll live with it. Make up numbers that match the condition and figure out what you do with the numbers and then do the same thing with the letters carefully, right? Physics, excuse my physics, none of you are my physics. Physics, I mean, we've been doing that all the time for a lot of those using principles of physics. So that we picked up that skill, hopefully. Does that make sense, Doug? Oh, I should circle the answer. What do I, A, C, D up plus A. The only thing I don't like about this, Martin, I like, I wish they would have written D plus A. This is one time I don't go alphabetically because I say vertical displacement plus the amplitude. I think D plus A mentally is how my brain does it but whatever, I can spot the twist there. Next, 20, 37? Good question. I sort of like this one. What do they want us to find here, Leslie? I don't see sine pi plus theta. I see that's an addition identity from my, I saw you glance there, right? That's alpha plus beta. So can you read to me the right-hand side but instead of alpha, can you say pi? And instead of beta, can you say theta? Go. Now I haven't used this yet. I'll come back to that M comma M. But by the way, is that an X? Is that a Y? How much you want to bet we're gonna be pulling out X, Y and R along the way as well? I'm putting that in my back pocket but I'm coming out with that. What's the sign of pi? By the way, if you haven't figured it out, I've always during this unit been saying, behold the human unit circle, you have your calculators on this test. What's the sign of pi? Is it? Like you can always type it in, right? What's the sign of pi? If you forget, if you want to check your answer, because you have your calculators this time, what's the sign of pi? So what's zero times cos theta? That whole thing vanishes. What's the cos of pi? Now you can get there also because Jesse, behold the human unit circle, let's clue in, you can cheat is the wrong word. You can be clever and use a crutch and maybe even turn that crutch into a bit of a stretcher almost on some questions on this test. What'd you say cos of pi was? So right now I get this negative sign theta. Leslie, how much you want to bet based on this now that they want me to go negative y over r? Right? I mean I haven't used the x, y thing and I put that in my back pocket but I've got signed by itself with a negative in front of it. What's y? What's sign? No, no, no, what's sign y over r with a negative in front of it? So you know what, I think, now how did I know to do that, Doug? Because I haven't used the x and y they gave me. What's y? N, negative. And by the way, wrong, wrong. And in fact I'm almost certain wrong because it's not negative but let's convince ourself that r apparently is the square root of m squared plus n squared. Oh, is r the square root of x squared plus y squared? Is r the square root of x squared plus y squared? Yeah, you know what? I'm pretty sure r is the square root of x squared plus y squared. Holy smokes. I didn't really need to do much of this nasty question. I was able to ditch that pretty quickly. I was able to ditch that pretty quickly and there was my clue to use y and r. Now, here's the other way they could have done this. Instead of giving it to you this way, they could have said the terminal, they could have said blah, blah, blah, blah, blah, blah, they could have said theta, let's say cos theta. Better yet, Mr. Newick, let's use x, y, and r. They could have said tan theta equals m over n. So the other way for them to give you x and y and r is to give you a trig function as a fraction. Oh, ooh, ooh. What if it wasn't a fraction? What's x in this case? One, okay, there's your complete hint. No, no, no, no, no. There's a negative there. There's just a negative there. What's sine y over r? So the negative is still there, right? What's y? No, what's y? There is no negative canceling out. That negative has just been this guy sitting in front. I also know that because they told me m and n are both positive over here too, okay? There is no two negatives. This negative is that negative is that negative, and then I filled in the y and I filled in the r, but I went over here. Oh, I guess there's a negative picking around on my answer. It can't be those two. Is that okay? Is Deena learning to look at the answers along the way and cross stuff out a lot? Oh, that would make my day. Cause you'll notice several times now, I haven't had to come close to finishing a question. I could get rid of all the wrong answers and we're good. Right? We're giggly now. What are we giggly now? Your gum stuck to your lip. Oh, when I was in college, and this was of course back in the late 80s, back in the fairly frizzy hair days, my friend was, his girlfriend had very frizzy hair. And we were sitting at the cafeteria tables and he took his gum out and kind of tossed it. And he thought he saw it land on the ground. She found it four days later in her hair, in a big hair ball. Oh, she had frizzy hair. Four days later. Yeah. Alfred. Hey, it was the 80s. Watch some of those hair bands and you'll get an idea of what it was like. And no, I never had hair like that. I kept, mine has been pretty short most of my life. Oh no, no, no. Yes. 95. Woo. I think that was a curve ball that I said, ah, yeah. B plus, A minus. What's an A plus question, Mr. Dewick? Every third or fourth provincial, they don't come out all that often, but an A plus question is where I gotta think too. Read to me. I agree. Keep going. I glance at my answers. I don't see any squares. Do you? Okay, so maybe, I don't think I'm gonna be trying to use like a sine squared plus cos squared equals one jump cause those have squares in them no matter what. Oh, I do notice the X is not by itself. What else is there in front of the X sitting there in my original question? Six. Okay, this is how I would handle this. I notice one more thing. So here's what I see a sine squared here with a minus sign in front of it. Yes? Somewhere on your formula sheet, can you find a sine squared with a minus sign in front of it? Okay, I, right now, here Doug, he said, I think it's double angle. Right now, I don't know where I'm going, but I'm leaning towards double angle and the first thing that I would write then is this. Well, I know that cosine of two theta equals, can you read to me the cos two theta answer that has a minus sign squared, the whole thing please. Oh, except in my example here, what's sitting where the theta is in my example here? Okay, now that you've got it all together in the correct order in one phrase, are you ready Doug? What's sitting where the theta is over here? I would probably just make a little note like this and I would say, I think that's a six X, like I think later on, wherever I see a theta, you know what I'm gonna put? Okay, what numbers do I have here and here? What is your GCF? If I pull a four out, what will that become? Ah, what will that become? Ah, what's in front of two of my answers? Ah, right, I might cross out A and B, but I'm not quite sure yet. Okay, you see I'm putting all this together though. This is a tough question, but I'm pulling out all the stops and I'm being stubborn and I would say, you know what? This is four bracket one minus two sine squared six X. Yes? Which is four times, now you're ready, here's our final move. What's one minus two sine squared? No, read it properly. Okay, ready, we gotta be really careful if you say it wrong, we're gonna end up in a nightmare here. What's this the same as? Except what did I say theta was? Right, in other words, I think this, if I try and plug it into there, where six X is theta, what, when you see it? And it is because of all of the different twos and I find on all of these for me, the toughest is going backwards from a double angle. In other words, if they give me cos two theta in disguise and I'm going backwards, that I find the toughest. Now Doug, you've finished the test. You're not sure about number 95. What kind of an idiot teacher gives you a test with 95 questions on it anyways? But that's okay, you like Mr. Dewick, you understand he makes mistakes occasionally, but I finished the test, I'm gonna go back and check number 95. What am I gonna do? No, not a math test. Well, the other stuff is factual recognition. Yeah, but no written section. Did you guys? Okay, fair enough. Okay, well that's a final. I'm talking a test, like a test. Look up, this is what I would do. Clear, clear. I would graph four minus eight. Now how do I do sine squared, bracket, sine eight X, close off the sine, close off, do I have the sine eight X all in brackets? Squared, that's how I type that. And I would be really lazy, I would go zoom, trig, I'm gonna rush. I would go mode, oh I am in radians, I would double check, did I type this in right? Four minus eight, oh sine eight X Mr. Dewick, six X squared. And the other thing, I look at my windows, they're going from negative two pi to two pi, I'm just gonna go from zero to two pi. It's ugly, I don't care. It looks like that. I would sketch that, oh you know what I would actually do? Turn this graph off, but leave it in. You go on the equal sign and press enter, that toggled it off. You see how now the equal sign is not blocked in anymore? Then I would go cos 12 X. That's not the right graph, is it? I would cross out A. Then I would go, maybe it's two cos six X. I'm trying B right now. That's not the right graph. I mean, I don't have to write this. I can remember what the ugly, that first thing looked like. Cross off B. Then I would go four cos six X. I can't remember what the first one looked like. Oh, you know what I'll do? I'll turn it back on, if they're the same graph, how many graphs should I actually see on my screen? Ah, they're not the same. I already know it's D probably, but I would do the same procedure with D. In other words, for a yucky identity, you don't hand this test in early until you've done every graphing calculator trick you can as well. Now, this is slow compared to the algebraic way because this would take me about four or five minutes. For algebraically, honestly, I would do it in about 60 seconds. So this is not a time saver, which is why I don't teach you this as the main and only method. And also because if you're planning and writing the provincial, this question here would be on the non-calc section. Most of the identity stuff would be. That's okay. You really have this loophole. I have to live with it. Besides, I'm putting three identities on the written and those ones you graphing calculator won't help you with because you've got to do the t-table algebraic proof thing anyways. Any others? Number four, written or multiple choice? Love to. What's weird about this? I don't have a sign by itself. Get the sign by itself. And then, so we, oh, sorry, let's write that because I can see we're tired. You'd have two sine x equals negative one. Now what? Yeah, I know you're tired, you're tripping on your brain. And that's why Martin was like, what? You're tripping on your brain. It's now you see, it's actually sine x, and you can see my pen is dying too, equals negative one half. And now it's gonna be cast rule. Sign is negative here and here. One, two, root, three triangle. Like I think the reference angle is pi by six. And so this is wrong. Oh, you know what? Hang on, stop, stop, stop, stop, stop. Can that be the correct answer? What quadrant is that in? What quadrant is that in? Did you hear me already say for a lot of these, you don't actually need to do the work. You can just do part of the question and figure out by looking at your answers and using a process of elimination. I'm telling you, when I've done up to number 75, I bet you one in five questions I didn't need to do, especially stuff like this cast rule, and you guys can do the cast rule in your sleep, right? There you go. Didn't even need to solve it, and I wouldn't. Why? Oh, because this test, I don't hand in early. The more extra time I have at the end, the more cheating I can do. I could graph this one and find it using my graphing calculator, right? The more being clever and using the technology I can do. Make sense? I'm willing to go tell five or we can knock off whenever, but I'm mentally, I got about 20 more minutes in me if we need it. And if you want me to print this up for you, I can. I will be uploading this to YouTube as well. Now I'm gonna hit upload and then go to the basketball game, which means I won't get back to your tell about 830 to send the email link out to people, but for my block H's, that won't be a big problem. And I think most of your block H anyways, only a couple of you aren't. Sorry. Next. Yes? I think five was weird. Oh, oh. Okay, so here I gave you the question. There's gonna be one like this where I give you the actual equation. I gave you the question. You always give us the question, sorry. I gave you the equation. I would read it carefully. D is depth of water. T is time. It's during hours. So first thing I'm gonna carefully type this in. Clear what I have. Three point, let's try that again. Three point four. Sine two pi bracket X minus. Do I need to go 7.00 or can I just type seven? Okay, good. Oh, don't forget to divide it by. By the way, can you see the period of this is 10.6? Cause it's two pi over B. I guess this tide is every 10.6 hours. Tides aren't exactly 12 hours. I think the first ones I did with you were exactly 12. They range a little bit, which is why the tides just don't repeat all year long, every time, every day. Close bracket, plus 2.8. Let's see, can I pick on you? Cause I think you were having trouble with windows. I'm gonna pick on you to help me with the window. Press window. What letter is sitting where the X normally does? G, time. What are we measuring time in? So I'm thinking 24 hours, probably one day, but I'm gonna double check here. Yeah, I better make sure I go at least past 6.30 p.m. So I'm gonna go from zero to 24. Good scale? Sure. If this had been a really nice period, I might have tried to pick a scale that went with that period, but I don't wanna go by 10.6 is yuck, so forget it. Y value is water depth. I like to see the ground. And then, Doug, are you ready? What's the highest this graph will get? No, look up. What's the highest this graph will get? 2.8 is the, and then plus 3.4. You know what? I'm gonna make the math easy. Three plus 3.4, roughly 6.4. I'm gonna go to eight, which is a nice kind of roundish number nearby there. See how I put that together? Scale? I think one actually will work. I don't answer. Graph. Tide goes down. Tide goes up. Now this is a bit of a yucky one. I don't like this because they have the tide going below the ground. Although here's what they would say. It's a seaport, they've dredged it. Okay, fine. I don't like the fact that I don't see the ground. So you know what I'm gonna do? View window. How about negative two? Ah, that looks better. There's no right window. There are wrong windows because you gotta be able to see the information. Is that okay, Leslie? Yep, bring it up. Okay, so when you hit graph and you get the syntax error, always pick option two, which stands for go to the error. Watch. Where did it put the cursor? You got an extra bracket. Don't, yeah, when you get the syntax error, that go to is really handy. Now let's see. Hey, there we go. Extra bracket. You guys get a syntax error and it has you option one quit. Option two, go to pick option two. It'll put the cursor right where the computer crashed, which usually is obvious when you see it there. Ah. Ready, Leslie? 6.30 p.m. Now, is that an X value or is that a Y value if they give me time? X, so I'm gonna use trace. Now here are the common mistakes. The first mistake would be a kid goes, oh, 6.3, no. What's 30 minutes as a decimal? So 6.5 would be the second mistake, no, p.m. And how much you wanna bet they have 6.5 as an answer there and 6.3 as an answer there. And 18.3 as an answer there. I'll bet you those are the four answers. D, right? There's gonna be a multiple choice where I give you the equation and say do something with it. Are we wrapping up? I'll do a couple more. Okay. Okay. Do you wanna print out or are you just watching anyways? I'll save this as a PDF and stick it somewhere. Okay, I've done a video as well and that's for the people that weren't here, but you guys weren't here. So give me one second. Right click.